Vibration damping, energy and energy flow in rods and beams: Governing formulae and semi-infinite systems

Generic features of energy and energy flow in thin rods and beams are investigated. Full equations of energy density and energy flow are formulated in terms of wave amplitudes. The differential equations for energy and energy flow are also formulated which are similar but not identical to some found earlier. The relationship between the power input and global kinetic and potential energies has been applied to rods and beams which gives easy access to their global energy. The present study is focused on semi-infinite rods and beams, and in particular, at the distribution of kinetic and potential energies within the section between the excitation and the end positions. A study of a finite beam system is presented in a companion paper. Both force and moment-type excitations are considered. It is shown that long finite beams closely match equivalent semi-infinite beams, where frequency band-averaged energy characteristics are concerned. Similar equivalence with infinite beams was found not to hold. It is further demonstrated that, unlike the kinetic energy density, the potential energy density exhibits a jump at the excitation point of a rod or a moment-driven beam. The damping was shown to reduce the jump in potential energy across the excitation point, but in turn increases the frequency range of strong power input to the end section.